There can surely no clichés about this being merely half-time. On a memorable night in Tallinn Giovanni Trapattoni's Republic of Ireland stepped decisively towards the finals of Euro 2012 with a performance that showed icy professionalism and ruthless opportunism.

Estonia proved to be limited opponents, their shortcomings particularly clear at the back, where they lost the centre-half Andrei Stepanov to a second booking in the 34th minute and the other centre-half Raio Piiroja to two more yellows received within six minutes of each other late in the second half. Yet Ireland twisted the knife, haunted perhaps by memories of past play-off nightmares, especially the one at the hand of Thierry Henry's France in the World Cup campaign two years ago.

They went a goal up early on courtesy of the outstanding Keith Andrews and plundered three more against tiring and increasingly punch-drunk opposition. Jon Walters' magnificent display got the goal it deserved and the captain Robbie Keane's 52nd and 53rd international strikes have surely ended the contest.

The celebrations on the Ireland bench after his first suggested they thought it was over at that point. Trapattoni, his staff and the substitutes leapt about like children and the final cut was provided by Keane's penalty, after the substitute Stephen Hunt had been foolishly tripped by another substitute, Ats Purje. His tackle was the half-hearted action of a beaten man and, all around him, Estonian chins were on the floor.

The home crowd turned their ire on the Hungarian referee Viktor Kassai – his decision to dismiss Piiroja for handball in the 76th minute particularly irked them – but they were deluding themselves if they thought he was to blame.

Ireland were thoroughly superior and the passion of the scenes at full-time told their own story, even if it looked incongruous to see the Football Association of Ireland's chief executive, John Delaney, in his suit, running about the pitch and punching the air like one of the players. The second-leg in Dublin has taken on the appearance of a formality.

The pitch was hardly pristine and Ireland had to acclimatise quickly during frenetic opening exchanges, when the pace and helter-skelter action gave the occasion a real cup tie feel. Estonia brimmed with almost manic exuberance but, if Trapattoni had hoped for cool calculation, he got it in the move that ended in Andrews heading the first goal.

Walters offered a passable impersonation of a raging bull throughout and was involved in the build-up, passing to Keane who supplied Aiden McGeady, and the winger's cross was met superbly by the onrushing Andrews.

The tie swung further in Ireland's favour when Estonia were reduced to 10 men. The home team's defence had been undermined by naivety and it was in evidence when Stepanov, already on a yellow card for a foul on McGeady and having been played into trouble, lunged for the ball only to be beaten to it by Keane. As the last man, when he clattered into the Ireland captain, there could be only one outcome.

Estonia threw what they had at Ireland early in the second half but they were ragged, their game undermined by sloppy individual errors. Ireland's streetwise control was impressive and they punched hard on the counter to assume a grip on the tie. McGeady was again at the heart of the move, his jink and shot forcing the goalkeeper Sergei Pareiko to parry. Keane pounced on the rebound, crossing deep to Walters who headed home at the far post.

Moments later Piiroja was booked for a foul and Andrew's viciously swerving free-kick perplexed Pareiko, who could only push the ball out to Keane, who made it three.

The crowd turned on Kassai when he dismissed Piiroja and there was yet another sting for Estonia in the booking for Pareiko, which leaves him suspended for the trip to Dublin. Trapattoni might consider playing himself.

Estonia finished at sixes and sevens, which felt harsh given the romanticism of their campaign when they have proved to be one of the most enjoyable surprise teams. The penalty was the final insult but Keane had no qualms about extracting the full punishment. Ireland can start planning for Poland and Ukraine.